Not and answer, but a long comment: I don't think a uniform rectangular grid works. Take a planar poisson process with intensity 1 and look at (0,n)x(0,n). This is your setup except I have replaced the uniforms with a poisson process with an expected n^2 events in the space. Divide it into a rectangular grid with (cn)^2 rectangles of area 1/c^2 each. The number of points in each rectangle is poisson with parameter 1/c^2,and different rectangles are independent. You wonder if any have of the (cn)^2 rectangles have two points in them. As the parameter is fixed there is a probability going to 1 that at least one of them does.
mike
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